The SE(3) Lie group framework for flexible multibody systems with contact

The kinematic description of multibody systems makes extensive use of the notion of
frames. Frame operations may be described in a systematic manner using concepts from
differential geometry and Lie groups, where a frame transformation is represented by an
element of the special Euclidean group SE(3). Working with left invariant derivatives
and a consistent spatial discretization leads to equations of motion formulated on a Lie
group. Forces, strain measures, arbitrary virtual motions and velocities are expressed in
the local body-attached frame such that the equations only depend on relative motions
between frames [1, 2]. Kinematic joints i.e., restricted relative motion modeled as bilateral
constraints [3], can be handled conveniently. Indeed, the SE(3) element that describes
relative transformations is invariant under superimposed Euclidean transformation. As
it will be shown in this contribution, the same can be said for contact conditions written
as unilateral constraints and the associated constraint gradient. The constraints are en-
forced using an augmented Lagrangian approach. An implicit Lie group time integration
scheme is employed [4]. The mass matrices and tangent stiffness contributions of each
element required for the semismooth Newton algorithm are invariant under rigid body
motions. Interestingly, during the entire formulation and discretization procedure, no
global parametrization of rotation is introduced and the non-linearity of the equations is
reduced as opposed to other formulations.